Generation and direct viscous dissipation of turbulent energy

Existing information about the generation and viscous dissipation of turbulent energy is based, as a rule, on the Laufer test data obtained for fluid flow in circular tubes at two Reynolds numbers (5 · 10⁵ and 5 · 10⁴). Computational dependences are presented herein for the generation and viscous dissipation of turbulent energy, common over the whole stream section and for the whole range of variation of the Reynolds number. The equation of the average energy balance during fluid flow in a circular tube and a flat channel is solved taking account of the equation of motion and the turbulent friction profile obtained by the author [1]. The computational dependences satisfy all the evident boundary conditions, agree with the Laufer test results [2] and yield a well-founded passage to the limit modes of average turbulent motion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–36, November–December, 1973.     

Mathematical modeling of motion of a fluid partially filling a rotating cylinder with radially aligned edges

A boundary-value problem of unsteady vortex motion of a viscous incompressible fluid with a free surface in a cavity rotating with a variable angular velocity and shaped as a straight circular cylinder with equidistant radial ribs is solved in a two-dimensional formulation by finite-difference methods. The drag coefficient of the rib is obtained as a function of its depth relative to the free surface.     

Instantaneous Squeeze-Film Force Between a Heat Exchanger Tube and a Support Plate for Arbitrary Tube Motion

The instantaneous squeeze-film force between a heat exchanger tube and a support plate is studied. Based on a two-dimensional rectangular plate model, a short-sleeve squeeze-film model for arbitrary tube motion is developed. The instantaneous squeeze-film force is expressed in normal and tangential directions. The normal squeeze-film force consists of four nonlinear terms, the viscous, unsteady inertia, convective inertia and centripetal inertia terms. Three nonlinear terms, the viscous, unsteady inertia and Coriolis inertia terms, make up the tangential squeeze-film force. An experimental apparatus was developed in order to evaluate the theoretical models against measurements of a finite length squeeze film. A modified model based on the experimental data is obtained where the viscous terms for both directions are multiplied by the instantaneous Reynolds number. All the inertia terms are multiplied by constant coefficients. The modified model is in good agreement with most experimental cases for unsymmetrical linear motion, approximate circular motion and elliptical motion. The form of the modified model is suitable for predicting instantaneous squeeze-film forces in the simulation of heat exchanger tube vibration. Further work using different sized components and fluid properties is required in order to finalize coefficient values.     

Dusty fluid round a cylinder in oscillatory rotation

Summary In this paper the problem is treated of the motion and number density of particles, initially at rest, under gravity in an incompressible, viscous fluid round a vertical, solid circular cylinder in oscillatory rotation. Explicit expressions are found for the particle velocities under certain conditions, the limitations of which are discussed. Methods are shown for finding trajectories and number densities of the particles. Simple asymptotic expressions are derived.     

Linearized equations of motion of a viscous fluid in a porous medium of periodic structure

The investigation of flow in essentially inhomogeneous porous systems through the analysis of model periodic structures [1] is considered. In the acoustic approximation, an integrodifferential equation is obtained that describes the motion of a viscous fluid in a rigid porous medium of periodic structure. The velocity vector and pressure are represented in the form of asymptotic series with respect to a small parameter that characterizes the size of the periodicity cell, and the well-known procedure for averaging linearized hydrodynamic equations with small coefficients of viscosity [2, 3] is also used. A solution is presented to the local problem in the periodicity cell for a structure consisting of a doubly periodic system of infinitely long rods of circular section and a compressible viscous fluid that fills the space between them, and also for a structure formed by a system of orthogonal rectilinear channels, filled with viscous fluid, in a solid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 123–130, March–April, 1988.     

A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid

We consider the gravity-driven motion of a heavy two-dimensional rigid body freely falling in a viscous fluid. We introduce a quasi-static linear model of the forces and torques induced by the possible changes in the body velocity, or by the occurrence of a nonzero incidence angle or a spanwise rotation of the body. The coefficients involved in this model are accurately computed over a full range of Reynolds number by numerically resolving the Navier–Stokes equations, considering three elementary situations where the motion of the body is prescribed. The falling body is found to exhibit three distinct eigenmodes which are always damped in the case of a thin plate with uniform mass loading or a circular cylinder, but may be amplified for other geometries, such as in the case of a square cylinder.     

Effect of the rotation,magnetic field and initial stress on peristaltic motion of micropolar fluid

In this work, the effect of magnetic field, rotation and initial stress on peristaltic motion of micropolar fluid in a circular cylindrical flexible tube with viscoelastic or elastic wall properties has been considered. Runge–Kutta technique are used. Runge–Kutta method is developed to solve the governing equations of motion resulting from a perturbation technique for small values of amplitude ratio. The time mean axial velocity profiles are presented for the case of free pumping and analyzed to observe the influence of wall properties, magnetic field, rotation and initial stress for various values of micropolar fluid parameters. In the case of viscoelastic wall, the effect of viscous damping on mean flow reversal at the boundary is seen. The numerical results of the time mean velocity profile are discussed in detail for homogeneous fluid under the effect of wall properties, magnetic field, initial stress and rotation for different cases by figures. The results indicate that the effect of wall properties, rotation, initial stress and magnetic field are very pronounced. Numerical results are given and illustrated graphically.     

Nonstationary expulsion of a viscous fluid from a pipe

The problem of expulsion of one viscous fluid by another in a circular pipe with a nonstationary pulsating laminar motion is considered. It is found that the imposition of pulsations of a definite frequency on the process of stationary expulsion can result in an increase in the coefficient of expulsion to 10–12%.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 170–171, March–April, 1977.     

Conjugate heat exchange and hydrodynamics for a viscous incompressible fluid moving in a rectangular cavity

Numerical simulation was performed of the motion of a viscous incompressible nonisothermal fluid in an open rectangular cavity under conditions of forced convection and conjugate heat exchange. The effect of the jet dynamic parameter (Reynolds number) and fluid flow conditions on the character of motion and heat exchange of viscous incompressible nonisothermal fluids in rectangular cavities is studied. A hydrodynamic pattern of viscous flow in an open cavity under forced convection conditions (in the conjugate and nonconjugate formulations of the problem) is obtained. The effect of parameters of the model on the character of motion is studied. Temperature profiles for the solid and fluid phases are obtained. The effect of parameters of the model on the character of temperature distribution in both phases is studied.     

Vibrations of a body in a bounded volume of viscous fluid

The problem of the vibrations of a body in a bounded volume of viscous fluid has been studied on a number of occasions [1–4]. The main attention has been devoted to determining the hydrodynamic characteristics of elements in the form of rods. Analytic solution of the problem is possible only in the simplest cases [2]. In the present paper, in which large Reynolds numbers are considered, the asymptotic method of Vishik and Lyusternik [5] and Chernous' ko [6] is used to consider the general problem of translational vibrations of an axisymmetric body in an axisymmetric volume of fluid. Equations of motion of the body and expressions for the coefficients due to the viscosity of the fluid are obtained. It is shown that in the first approximation these coefficients differ only by a constant factor and are completely determined if the solution to the problem for an ideal fluid is known. Examples are given of the determination of the “viscous” added mass and the damping coefficient for some bodies and cavities. In the case of an ideal fluid, general estimates are obtained for the added mass and also for the influence of nonlinearity. Ritz's method is used to solve the problem of longitudinal vibrations of an ellipsoid of revolution in a circular cylinder. The hydrodynamic coefficients have been determined numerically on a computer. The theoretical results agree well with the results of experimental investigations.     

Displacement of a viscous fluid from a Hele-Shaw cell with a sink

It is found that the radial geometry does not stabilize the evolution of instability in the displacement of a more viscous fluid by a less viscous fluid from a circular Hele-Shaw cell with a sink. A linear analysis shows the absolute instability of the radial displacement front. The appearance of isolated fingers is observed during numerical simulations.     

Wave propagation in a fluid flowing through a curved thin-walled elastic tube

The pulsatile flow in a curved elastic pipe of circular cross section is investigated. The unsteady flow of a viscous fluid and the wall motion equations are written in a toroidal coordinate system, superimposed and linearized over a steady state solution. Being the main application relative to the vascular system, the radius of the pipe is assumed small compared with the radius of curvature. This allows an asymptotic analysis over the curvature parameter. The model results an extension of the Womersley's model for the straight elastic tube. A numerical solution is found for the first order approximation and computational results are finally presented, demonstrating the role of curvature in the wave propagation and in the development of a secondary flow.     

Slow motions of a circular cylinder experiencing slip near a plane wall

A theoretical study is presented for the two-dimensional creeping flow caused by a long circular cylindrical particle translating and rotating in a viscous fluid near a large plane wall parallel to its axis. The fluid is allowed to slip at the surface of the particle. The Stokes equations for the fluid velocity field are solved in the quasi-steady limit using cylindrical bipolar coordinates. Semi-analytical solutions for the drag force and torque acting on the particle by the fluid are obtained for various values of the slip coefficient associated with the particle surface and of the relative separation distance between the particle and the wall. The results indicate that the translation and rotation of the confined cylinder are not coupled with each other. For the motion of a no-slip cylinder near a plane wall, our hydrodynamic drag force and torque results reduce to the closed-form solutions available in the literature. The boundary-corrected drag force and torque acting on the particle decrease with an increase in the slip coefficient for an otherwise specified condition. The plane wall exerts the greatest drag on the particle when its migration occurs normal to it, and the least in the case of motion parallel to it. The enhancement in the hydrodynamic drag force and torque on a translating and rotating particle caused by a nearby plane wall is much more significant for a cylinder than for a sphere.     

A viscous vortex particle method for deforming bodies with application to biolocomotion

Bio‐inspired mechanics of locomotion generally consist of the interaction of flexible structures with the surrounding fluid to generate propulsive forces. In this work, we extend, for the first time, the viscous vortex particle method (VVPM) to continuously deforming two‐dimensional bodies. The VVPM is a high‐fidelity Navier–Stokes computational method that captures the fluid motion through evolution of vorticity‐bearing computational particles. The kinematics of the deforming body surface are accounted for via a surface integral in the Biot–Savart velocity. The spurious slip velocity in each time step is removed by computing an equivalent vortex sheet and allowing it to flux to adjacent particles; hence, no‐slip boundary conditions are enforced. Particles of both uniform and variable size are utilized, and their relative merits are considered. The placement of this method in the larger class of immersed boundary methods is explored. Validation of the method is carried out on the problem of a periodically deforming circular cylinder immersed in a stagnant fluid, for which an analytical solution exists when the deformations are small. We show that the computed vorticity and velocity of this motion are both in excellent agreement with the analytical solution. Finally, we explore the fluid dynamics of a simple fish‐like shape undergoing undulatory motion when immersed in a uniform free stream, to demonstrate the application of the method to investigations of biomorphic locomotion. Copyright © 2008 John Wiley & Sons, Ltd.     

Structure of the drift flow initiated by periodic waves traveling over a viscous fluid surface

An analytical procedure used in calculating the Stokes drift velocity (the drift motion initiated by the propagation of a capillary-gravity wave over an ideal fluid surface) is applied to the problem of the calculation of an analogous drift flow in a viscous fluid. An expression for the velocity of the Stokes drift modified with allowance for viscosity is constructed. The properties and the role of the modified Stokes drift in the general pattern of the drift in a viscous fluid are analyzed.     

Motion of a solidcontaininga spherical damper

For the purpose of modeling the motion of a solid with a cavity filled with a viscous fluid, M. A. Lavrent'ev [1] has proposed a model in the form of a solid with a spherical cavity in which another solid, spherical in shape, is enclosed. The sphere is separated from the cavity walls by a small, clearance in which viscous forces act (a lubricating film). This simple model with a finite number of degrees of freedom possesses certain mechanical properties of a solid with a cavity containing a viscous fluid. Study of this model is therefore of interest.The present paper examines certain properties of the model, which will be termed a solid with a damper. It is shown that for a highviscosity lubricant the motion of a solid with a damper can be described by the same equations which pertain to the motion of a solid with a spherical cavity filled with a high-viscosity fluid. Expressions relating the parameters of the systems are obtained. If these relations are fulfilled, the systems become mechanically equivalent.The steady motions of a free solid with a damper and their stability conditions are determined.These motions and stability conditions hold for a body with a cavity filled with a viscous fluid [2].     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

Unsteady motion of viscous fluid in an expanding tube

The problem of unsteady motion of a viscous compressible fluid in a semiinfinite tube with horizontal axis is solved by successive approximation. The circular cross section of the tube depends exponentially on the coordinate measured along the tube axis. At the end of the tube there is a unit such as a sliding valve, compressor, reciprocating pump, or turbine that changes the flow rate. The process is assumed to be barotropic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 79–82, May–June, 1983.     

Steady laminar flow of viscoelastic fluid in a curved pipe of circular cross-section with varying curvature

The steady laminar flow of viscoelastic fluid through pipes of circular cross-section, whose center-line curvature varies locally, is analyzed theoretically. The flows in three kinds of pipes whose center-lines are specified by

as the examples of once, twice, and periodically curved pipes, respectively, are discussed in comparison with purely viscous flow. The analysis is valid for any other two-dimensionally curved pipes, when the center-line curvature is small. In addition, the reason why the secondary flow of a viscoelastic fluid in a curved pipe of circular cross-section is stronger than that of a purely viscous fluid is explained. In the present paper, the White—Metzner model is employed as the constitutive equation.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.